Intuitionistic logic

Roughly speaking, 'intuitionism' holds that logic and mathematics are 'constructive' mental activities. That is, they are not analytic activities wherein deep properties of existence are revealed and applied. Instead, logic and mathematics are the application of internally consistent methods to realize more complex mental constructs (really, a kind of game). In a stricter sense, intuitionistic logic can be investigated as a very concrete and formal kind of mathematical logic. While it may be argued whether such a formal calculus really captures the philosophical aspects of intuitionism, it has properties which are also quite useful from a practical point of view.
Both notions of the term will be considered below.

As an example of this difference, law of the excluded middle, while classically valid, is not intuitionistically valid, because, in a logical calculus that allows it, it's possible to argue P ∨ ¬P without knowing which one specifically is the case.
This is fine if one assumes that the law of the excluded middle is some kind of subtle truth inherent in the nature of being; but if the validity of a mental construct is entirely dependent upon its coherence with its context (i.e., the mind), then epistemological opacity is, in effect, cheating. In intuitionistic logic, it is not permitted to assert a disjunction such as P ∨ ¬P without also being able to say specifically which one is true. More generally, the formula P ∨ ¬P is not a theorem of intuitionistic logic as it is of classical logic. In classical logic,
P ∨ ¬P means that one of P or ¬P is true; in
intuitionistic logic, P ∨ ¬P means that one of P or ¬P can be proved, which is a much stronger statement, and which might not always be the case.

Intuitionistic logic substitutes justification for truth in its logical calculus. Instead of a deterministic, bivalent truth assignment scheme, it allows for a third, indeterminate truth value. A proposition may be provably justified, or provably not justified, or undetermined. The logical calculus preserves justification, rather than truth, across transformations yielding derived propositions.

A more familiar example of a classical tautology which is invalid in intuitionistic logic concerns the so-called 'double negation elimination'.
In classical logic, both P → ¬¬P and also
¬¬P → P are theorems. In intuitionistic logic, only the first is a theorem---double negation can be introduced, but not eliminated.
The interpretation of negation in intuitionistic logic is different from its counterpart in classical logic. In classical logic, ¬P asserts that P is false; in intuitionistic logic, ¬P asserts that a proof of P is impossible. The asymmetry between the two implications above now becomes apparent. If P is provable, then it is certainly impossible to prove that there is no proof of P; this is the first implication. But the second implication fails: just because if there is no proof that a proof of P is impossible, we cannot conclude from this absence that there is a proof of P.

The observation that many classically valid tautologies are not theorems of
intuitionistic logic leads to the idea of weakening the proof theory of
classical logic. This has for example been done by Gentzen
with his sequent calculus LK, obtaining a weaker version,
that he called LJ. This gives a suitable proof theory.

The semantics are rather more complicated than for the classical, two-valued case. A model theory can be given by Heyting algebras or, equivalently, by Kripke semantics.

A valuation on propositional variables can be given by assigning elements of a Heyting algebra. The valuation can then be extended to formulae by matching the propositional connectives with their corresponding operations in the algebra. A valid sentence is then one which has valuation 1 in any valuation on any Heyting algebra.

It can be shown that we need in fact consider only the Heyting algebra given by the open sets of the real plane with its usual topology — intuitionistic validities correspond precisely to Heyting formulae which evaluate to the entire plane for any assignment of open subsets to the variables.

For example, the law of the excluded middle can then easily be seen not to be valid — let A be the strict upper plane, {(x,y) | y > 0}, then ¬A = Int(R^2 \\ A) = {(x,y) | y < 0}, the strict lower plane, so A ∨ ¬A = {(x,y) | y!=0} != R^2. So we do not have |= P ∨ ¬P in intuistionistic logic.